Algebraic Structures on Finite Sets

Abstract

In IB we introduced the characteristic functions c _S for subsets S of a set U and proved (IB2) that the function c _S ↦ S is a bijection between K^U and P(U). Subsequently it was to be verified (Exercise IB3) that this same bijection made the assignments c _S + c _T ↦ S + T and c _S c _T ↦ S ∩ T. We have thereby that (P(U), +, ∩) is “algebra-isomorphic” to the commutative algebra (K^U, +, ·), and hence (P(U), +, ∩) is a commutative algebra over the field K. In particular, (P(U), +) is a vector space over K, while (P(U), +, ∩) is a commutative ring; ∅ is the additive identity and U itself is the multiplicative identity. For the present we shall be concerned only with the vector space structure.